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Supervised Learning : Exploring Regularization Approaches for Controlling Model Complexity Through Weight Penalization for Neural Network Classification

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John Pauline Pineda

April 27, 2024

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• 1. Table of Contents
  • 1.1 Data Background
  • 1.2 Data Description
  • 1.3 Data Quality Assessment
  • 1.4 Data Preprocessing
    • 1.4.1 Data Cleaning
    • 1.4.2 Missing Data Imputation
    • 1.4.3 Outlier Treatment
    • 1.4.4 Collinearity
    • 1.4.5 Shape Transformation
    • 1.4.6 Centering and Scaling
    • 1.4.7 Data Encoding
    • 1.4.8 Preprocessed Data Description
  • 1.5 Data Exploration
    • 1.5.1 Exploratory Data Analysis
    • 1.5.2 Hypothesis Testing
  • 1.6 Neural Network Classification Gradient and Weight Updates
    • 1.6.1 Premodelling Data Description
    • 1.6.2 No Regularization
    • 1.6.3 L1 Regularization
    • 1.6.4 L2 Regularization
    • 1.6.5 ElasticNet Regularization
  • 1.7 Consolidated Findings
• 2. Summary
• 3. References

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1. Table of Contents

This project manually implements the L1 Regularization, L2 Regularization and Elasticnet Regularization algorithms using various helpful packages in Python to penalize large weights, reduce model complexity and improve generalization with fixed values applied for the learning rate and iteration count parameters to optimally update the gradients and weights of an artificial neural network classification model. The cost function, classification accuracy and layer weight optimization profiles of the different regularization algorithms were compared. All results were consolidated in a Summary presented at the end of the document.

Artificial Neural Network, in the context of categorical response prediction, consists of interconnected nodes called neurons organized in layers. The model architecture involves an input layer which receives the input data, with each neuron representing a feature or attribute of the data; hidden layers which perform computations on the input data through weighted connections between neurons and apply activation functions to produce outputs; and the output layer which produces the final predictions equal to the number of classes, each representing the probability of the input belonging to a particular class, based on the computations performed in the hidden layers. Neurons within adjacent layers are connected by weighted connections. Each connection has an associated weight that determines the strength of influence one neuron has on another. These weights are adjusted during the training process to enable the network to learn from the input data and make accurate predictions. Activation functions introduce non-linearities into the network, allowing it to learn complex relationships between inputs and outputs. The training process involves presenting input data along with corresponding target outputs to the network and adjusting the weights to minimize the difference between the predicted outputs and the actual targets which is typically performed through optimization algorithms such as gradient descent and backpropagation. The training process iteratively updates the weights until the model's predictions closely match the target outputs.

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Regularization Algorithms, in the context of neural network classification, are techniques used to prevent overfitting and improve the generalization performance of the model by imposing constraints on its parameters during training. These constraints are typically applied to the weights of the neural network and are aimed at reducing model complexity, controlling the magnitude of the weights, and promoting simpler and more generalizable solutions. Regularization approaches work by adding penalty terms to the loss function during training. These penalty terms penalize large weights or complex models, encouraging the optimization process to prioritize simpler solutions that generalize well to unseen data. By doing so, regularization helps prevent the neural network from fitting noise or irrelevant patterns present in the training data and encourages it to learn more robust and meaningful representations.

1.1. Data Background

Datasets used for the analysis were separately gathered and consolidated from various sources including:

1. Cancer Rates from World Population Review
2. Social Protection and Labor Indicator from World Bank
3. Education Indicator from World Bank
4. Economy and Growth Indicator from World Bank
5. Environment Indicator from World Bank
6. Climate Change Indicator from World Bank
7. Agricultural and Rural Development Indicator from World Bank
8. Social Development Indicator from World Bank
9. Health Indicator from World Bank
10. Science and Technology Indicator from World Bank
11. Urban Development Indicator from World Bank
12. Human Development Indices from Human Development Reports
13. Environmental Performance Indices from Yale Center for Environmental Law and Policy

This study hypothesized that various global development indicators and indices influence cancer rates across countries.

The target variable for the study is:

• CANRAT - Dichotomized category based on age-standardized cancer rates, per 100K population (2022)

The predictor variables for the study are:

• GDPPER - GDP per person employed, current US Dollars (2020)
• URBPOP - Urban population, % of total population (2020)
• PATRES - Patent applications by residents, total count (2020)
• RNDGDP - Research and development expenditure, % of GDP (2020)
• POPGRO - Population growth, annual % (2020)
• LIFEXP - Life expectancy at birth, total in years (2020)
• TUBINC - Incidence of tuberculosis, per 100K population (2020)
• DTHCMD - Cause of death by communicable diseases and maternal, prenatal and nutrition conditions, % of total (2019)
• AGRLND - Agricultural land, % of land area (2020)
• GHGEMI - Total greenhouse gas emissions, kt of CO2 equivalent (2020)
• RELOUT - Renewable electricity output, % of total electricity output (2015)
• METEMI - Methane emissions, kt of CO2 equivalent (2020)
• FORARE - Forest area, % of land area (2020)
• CO2EMI - CO2 emissions, metric tons per capita (2020)
• PM2EXP - PM2.5 air pollution, population exposed to levels exceeding WHO guideline value, % of total (2017)
• POPDEN - Population density, people per sq. km of land area (2020)
• GDPCAP - GDP per capita, current US Dollars (2020)
• ENRTER - Tertiary school enrollment, % gross (2020)
• HDICAT - Human development index, ordered category (2020)
• EPISCO - Environment performance index , score (2022)

1.2. Data Description

1. The dataset is comprised of:
   • 177 rows (observations)
   • 22 columns (variables)
     • 1/22 metadata (object)
       • COUNTRY
     • 1/22 target (categorical)
       • CANRAT
     • 19/22 predictor (numeric)
       • GDPPER
       • URBPOP
       • PATRES
       • RNDGDP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • RELOUT
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • ENRTER
       • EPISCO
     • 1/22 predictor (categorical)
       • HDICAT

##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
%matplotlib inline

from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import StandardScaler
from scipy import stats

##################################
# Loading the dataset
##################################
cancer_rate = pd.read_csv('CategoricalCancerRates.csv')

##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)

Column Names and Data Types:

COUNTRY     object
CANRAT      object
GDPPER     float64
URBPOP     float64
PATRES     float64
RNDGDP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
RELOUT     float64
METEMI     float64
FORARE     float64
CO2EMI     float64
PM2EXP     float64
POPDEN     float64
ENRTER     float64
GDPCAP     float64
HDICAT      object
EPISCO     float64
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()

	COUNTRY	CANRAT	GDPPER	URBPOP	PATRES	RNDGDP	POPGRO	LIFEXP	TUBINC	DTHCMD	...	RELOUT	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	ENRTER	GDPCAP	HDICAT	EPISCO
0	Australia	High	98380.63601	86.241	2368.0	NaN	1.235701	83.200000	7.2	4.941054	...	13.637841	131484.763200	17.421315	14.772658	24.893584	3.335312	110.139221	51722.06900	VH	60.1
1	New Zealand	High	77541.76438	86.699	348.0	NaN	2.204789	82.256098	7.2	4.354730	...	80.081439	32241.937000	37.570126	6.160799	NaN	19.331586	75.734833	41760.59478	VH	56.7
2	Ireland	High	198405.87500	63.653	75.0	1.23244	1.029111	82.556098	5.3	5.684596	...	27.965408	15252.824630	11.351720	6.768228	0.274092	72.367281	74.680313	85420.19086	VH	57.4
3	United States	High	130941.63690	82.664	269586.0	3.42287	0.964348	76.980488	2.3	5.302060	...	13.228593	748241.402900	33.866926	13.032828	3.343170	36.240985	87.567657	63528.63430	VH	51.1
4	Denmark	High	113300.60110	88.116	1261.0	2.96873	0.291641	81.602439	4.1	6.826140	...	65.505925	7778.773921	15.711000	4.691237	56.914456	145.785100	82.664330	60915.42440	VH	77.9

5 rows × 22 columns

##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].astype('category')
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].cat.set_categories(['Low', 'High'], ordered=True)
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)

##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())

Numeric Variable Summary:

	count	mean	std	min	25%	50%	75%	max
GDPPER	165.0	45284.424283	3.941794e+04	1718.804896	13545.254510	34024.900890	66778.416050	2.346469e+05
URBPOP	174.0	59.788121	2.280640e+01	13.345000	42.432750	61.701500	79.186500	1.000000e+02
PATRES	108.0	20607.388889	1.340683e+05	1.000000	35.250000	244.500000	1297.750000	1.344817e+06
RNDGDP	74.0	1.197474	1.189956e+00	0.039770	0.256372	0.873660	1.608842	5.354510e+00
POPGRO	174.0	1.127028	1.197718e+00	-2.079337	0.236900	1.179959	2.031154	3.727101e+00
LIFEXP	174.0	71.746113	7.606209e+00	52.777000	65.907500	72.464610	77.523500	8.456000e+01
TUBINC	174.0	105.005862	1.367229e+02	0.770000	12.000000	44.500000	147.750000	5.920000e+02
DTHCMD	170.0	21.260521	1.927333e+01	1.283611	6.078009	12.456279	36.980457	6.520789e+01
AGRLND	174.0	38.793456	2.171551e+01	0.512821	20.130276	40.386649	54.013754	8.084112e+01
GHGEMI	170.0	259582.709895	1.118550e+06	179.725150	12527.487367	41009.275980	116482.578575	1.294287e+07
RELOUT	153.0	39.760036	3.191492e+01	0.000296	10.582691	32.381668	63.011450	1.000000e+02
METEMI	170.0	47876.133575	1.346611e+05	11.596147	3662.884908	11118.976025	32368.909040	1.186285e+06
FORARE	173.0	32.218177	2.312001e+01	0.008078	11.604388	31.509048	49.071780	9.741212e+01
CO2EMI	170.0	3.751097	4.606479e+00	0.032585	0.631924	2.298368	4.823496	3.172684e+01
PM2EXP	167.0	91.940595	2.206003e+01	0.274092	99.627134	100.000000	100.000000	1.000000e+02
POPDEN	174.0	200.886765	6.453834e+02	2.115134	27.454539	77.983133	153.993650	7.918951e+03
ENRTER	116.0	49.994997	2.970619e+01	2.432581	22.107195	53.392460	71.057467	1.433107e+02
GDPCAP	170.0	13992.095610	1.957954e+04	216.827417	1870.503029	5348.192875	17421.116227	1.173705e+05
EPISCO	165.0	42.946667	1.249086e+01	18.900000	33.000000	40.900000	50.500000	7.790000e+01

##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())

Object Variable Summary:

	count	unique	top	freq
COUNTRY	177	177	Australia	1

##################################
# Performing a general exploration of the categorical variables
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())

Categorical Variable Summary:

	count	unique	top	freq
CANRAT	177	2	Low	132
HDICAT	167	4	VH	59

1.3. Data Quality Assessment

Data quality findings based on assessment are as follows:

1. No duplicated rows observed.
2. Missing data noted for 20 variables with Null.Count>0 and Fill.Rate<1.0.
   • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
   • PATRES: Null.Count = 69, Fill.Rate = 0.610
   • ENRTER: Null.Count = 61, Fill.Rate = 0.655
   • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • GDPPER: Null.Count = 12, Fill.Rate = 0.932
   • EPISCO: Null.Count = 12, Fill.Rate = 0.932
   • HDICAT: Null.Count = 10, Fill.Rate = 0.943
   • PM2EXP: Null.Count = 10, Fill.Rate = 0.943
   • DTHCMD: Null.Count = 7, Fill.Rate = 0.960
   • METEMI: Null.Count = 7, Fill.Rate = 0.960
   • CO2EMI: Null.Count = 7, Fill.Rate = 0.960
   • GDPCAP: Null.Count = 7, Fill.Rate = 0.960
   • GHGEMI: Null.Count = 7, Fill.Rate = 0.960
   • FORARE: Null.Count = 4, Fill.Rate = 0.977
   • TUBINC: Null.Count = 3, Fill.Rate = 0.983
   • AGRLND: Null.Count = 3, Fill.Rate = 0.983
   • POPGRO: Null.Count = 3, Fill.Rate = 0.983
   • POPDEN: Null.Count = 3, Fill.Rate = 0.983
   • URBPOP: Null.Count = 3, Fill.Rate = 0.983
   • LIFEXP: Null.Count = 3, Fill.Rate = 0.983
3. 120 observations noted with at least 1 missing data. From this number, 14 observations reported high Missing.Rate>0.2.
   • COUNTRY=Guadeloupe: Missing.Rate= 0.909
   • COUNTRY=Martinique: Missing.Rate= 0.909
   • COUNTRY=French Guiana: Missing.Rate= 0.909
   • COUNTRY=New Caledonia: Missing.Rate= 0.500
   • COUNTRY=French Polynesia: Missing.Rate= 0.500
   • COUNTRY=Guam: Missing.Rate= 0.500
   • COUNTRY=Puerto Rico: Missing.Rate= 0.409
   • COUNTRY=North Korea: Missing.Rate= 0.227
   • COUNTRY=Somalia: Missing.Rate= 0.227
   • COUNTRY=South Sudan: Missing.Rate= 0.227
   • COUNTRY=Venezuela: Missing.Rate= 0.227
   • COUNTRY=Libya: Missing.Rate= 0.227
   • COUNTRY=Eritrea: Missing.Rate= 0.227
   • COUNTRY=Yemen: Missing.Rate= 0.227
4. Low variance observed for 1 variable with First.Second.Mode.Ratio>5.
   • PM2EXP: First.Second.Mode.Ratio = 53.000
5. No low variance observed for any variable with Unique.Count.Ratio>10.
6. High skewness observed for 5 variables with Skewness>3 or Skewness<(-3).
   • POPDEN: Skewness = +10.267
   • GHGEMI: Skewness = +9.496
   • PATRES: Skewness = +9.284
   • METEMI: Skewness = +5.801
   • PM2EXP: Skewness = -3.141

##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()

0

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                              data_type_list,
                                              row_count_list,
                                              non_null_count_list,
                                              null_count_list,
                                              fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(all_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	COUNTRY	object	177	177	0	1.000000
1	CANRAT	category	177	177	0	1.000000
2	GDPPER	float64	177	165	12	0.932203
3	URBPOP	float64	177	174	3	0.983051
4	PATRES	float64	177	108	69	0.610169
5	RNDGDP	float64	177	74	103	0.418079
6	POPGRO	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051
9	DTHCMD	float64	177	170	7	0.960452
10	AGRLND	float64	177	174	3	0.983051
11	GHGEMI	float64	177	170	7	0.960452
12	RELOUT	float64	177	153	24	0.864407
13	METEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
15	CO2EMI	float64	177	170	7	0.960452
16	PM2EXP	float64	177	167	10	0.943503
17	POPDEN	float64	177	174	3	0.983051
18	ENRTER	float64	177	116	61	0.655367
19	GDPCAP	float64	177	170	7	0.960452
20	HDICAT	category	177	167	10	0.943503
21	EPISCO	float64	177	165	12	0.932203

##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])

20

##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
5	RNDGDP	float64	177	74	103	0.418079
4	PATRES	float64	177	108	69	0.610169
18	ENRTER	float64	177	116	61	0.655367
12	RELOUT	float64	177	153	24	0.864407
2	GDPPER	float64	177	165	12	0.932203
21	EPISCO	float64	177	165	12	0.932203
20	HDICAT	category	177	167	10	0.943503
16	PM2EXP	float64	177	167	10	0.943503
9	DTHCMD	float64	177	170	7	0.960452
13	METEMI	float64	177	170	7	0.960452
15	CO2EMI	float64	177	170	7	0.960452
19	GDPCAP	float64	177	170	7	0.960452
11	GHGEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
8	TUBINC	float64	177	174	3	0.983051
10	AGRLND	float64	177	174	3	0.983051
6	POPGRO	float64	177	174	3	0.983051
17	POPDEN	float64	177	174	3	0.983051
3	URBPOP	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051

##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]

##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()

##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))

##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))

##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)

##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
                                           column_count_list,
                                           null_row_list,
                                           missing_rate_list), 
                                        columns=['Row.Name',
                                                 'Column.Count',
                                                 'Null.Count',                                                 
                                                 'Missing.Rate'])
display(all_row_quality_summary)

	Row.Name	Column.Count	Null.Count	Missing.Rate
0	Australia	22	1	0.045455
1	New Zealand	22	2	0.090909
2	Ireland	22	0	0.000000
3	United States	22	0	0.000000
4	Denmark	22	0	0.000000
...	...	...	...	...
172	Congo Republic	22	3	0.136364
173	Bhutan	22	2	0.090909
174	Nepal	22	2	0.090909
175	Gambia	22	4	0.181818
176	Niger	22	2	0.090909

177 rows × 4 columns

##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])

120

##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])

14

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))

	Row.Name	Column.Count	Null.Count	Missing.Rate
35	Guadeloupe	22	20	0.909091
39	Martinique	22	20	0.909091
56	French Guiana	22	20	0.909091
13	New Caledonia	22	11	0.500000
44	French Polynesia	22	11	0.500000
75	Guam	22	11	0.500000
53	Puerto Rico	22	9	0.409091
85	North Korea	22	6	0.272727
132	Somalia	22	6	0.272727
168	South Sudan	22	6	0.272727
73	Venezuela	22	5	0.227273
117	Libya	22	5	0.227273
161	Eritrea	22	5	0.227273
164	Yemen	22	5	0.227273

##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns

##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()

##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()

##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()

##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()

##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]

##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]

##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)

##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)

##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))

##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()

##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()

numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                numeric_minimum_list,
                                                numeric_mean_list,
                                                numeric_median_list,
                                                numeric_maximum_list,
                                                numeric_first_mode_list,
                                                numeric_second_mode_list,
                                                numeric_first_mode_count_list,
                                                numeric_second_mode_count_list,
                                                numeric_first_second_mode_ratio_list,
                                                numeric_unique_count_list,
                                                numeric_row_count_list,
                                                numeric_unique_count_ratio_list,
                                                numeric_skewness_list,
                                                numeric_kurtosis_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Minimum',
                                                 'Mean',
                                                 'Median',
                                                 'Maximum',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio',
                                                 'Skewness',
                                                 'Kurtosis'])
display(numeric_column_quality_summary)

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
0	GDPPER	1718.804896	45284.424283	34024.900890	2.346469e+05	98380.636010	42154.178100	1	1	1.000000	165	177	0.932203	1.517574	3.471992
1	URBPOP	13.345000	59.788121	61.701500	1.000000e+02	100.000000	52.516000	2	1	2.000000	173	177	0.977401	-0.210702	-0.962847
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
3	RNDGDP	0.039770	1.197474	0.873660	5.354510e+00	1.232440	0.962180	1	1	1.000000	74	177	0.418079	1.396742	1.695957
4	POPGRO	-2.079337	1.127028	1.179959	3.727101e+00	1.235701	1.483129	1	1	1.000000	174	177	0.983051	-0.195161	-0.423580
5	LIFEXP	52.777000	71.746113	72.464610	8.456000e+01	83.200000	68.687000	1	1	1.000000	174	177	0.983051	-0.357965	-0.649601
6	TUBINC	0.770000	105.005862	44.500000	5.920000e+02	12.000000	7.200000	4	3	1.333333	131	177	0.740113	1.746333	2.429368
7	DTHCMD	1.283611	21.260521	12.456279	6.520789e+01	4.941054	42.079403	1	1	1.000000	170	177	0.960452	0.900509	-0.691541
8	AGRLND	0.512821	38.793456	40.386649	8.084112e+01	46.252480	72.006469	1	1	1.000000	174	177	0.983051	0.074000	-0.926249
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	3000.932259	1	1	1.000000	170	177	0.960452	9.496120	101.637308
10	RELOUT	0.000296	39.760036	32.381668	1.000000e+02	100.000000	13.637841	3	1	3.000000	151	177	0.853107	0.501088	-0.981774
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	1326.034028	1	1	1.000000	170	177	0.960452	5.801014	38.661386
12	FORARE	0.008078	32.218177	31.509048	9.741212e+01	17.421315	8.782159	1	1	1.000000	173	177	0.977401	0.519277	-0.322589
13	CO2EMI	0.032585	3.751097	2.298368	3.172684e+01	14.772658	0.972088	1	1	1.000000	170	177	0.960452	2.721552	10.311574
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	13.300785	1	1	1.000000	174	177	0.983051	10.267750	119.995256
16	ENRTER	2.432581	49.994997	53.392460	1.433107e+02	110.139221	45.220661	1	1	1.000000	116	177	0.655367	0.275863	-0.392895
17	GDPCAP	216.827417	13992.095610	5348.192875	1.173705e+05	51722.069000	3961.726633	1	1	1.000000	170	177	0.960452	2.258568	5.938690
18	EPISCO	18.900000	42.946667	40.900000	7.790000e+01	29.600000	43.600000	3	3	1.000000	137	177	0.774011	0.641799	0.035208

##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])

1

##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
14	PM2EXP	0.274092	91.940595	100.0	100.0	100.0	100.0	106	2	53.0	61	177	0.344633	-3.141557	9.032386

##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])

5

##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	13.300785	1	1	1.000000	174	177	0.983051	10.267750	119.995256
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	3000.932259	1	1	1.000000	170	177	0.960452	9.496120	101.637308
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	1326.034028	1	1	1.000000	170	177	0.960452	5.801014	38.661386
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386

##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')

##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns

##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]

##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]

##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]

##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]

##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)

##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)

##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))

##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)

object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
                                                 object_first_mode_list,
                                                 object_second_mode_list,
                                                 object_first_mode_count_list,
                                                 object_second_mode_count_list,
                                                 object_first_second_mode_ratio_list,
                                                 object_unique_count_list,
                                                 object_row_count_list,
                                                 object_unique_count_ratio_list), 
                                        columns=['Object.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(object_column_quality_summary)

	Object.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	COUNTRY	Australia	Mauritius	1	1	1.0	177	177	1.0

##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')

##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns

##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]

##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]

##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)

##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)

##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)

categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
                                                    categorical_first_mode_list,
                                                    categorical_second_mode_list,
                                                    categorical_first_mode_count_list,
                                                    categorical_second_mode_count_list,
                                                    categorical_first_second_mode_ratio_list,
                                                    categorical_unique_count_list,
                                                    categorical_row_count_list,
                                                    categorical_unique_count_ratio_list), 
                                        columns=['Categorical.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(categorical_column_quality_summary)

	Categorical.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	CANRAT	Low	High	132	45	2.933333	2	177	0.011299
1	HDICAT	VH	H	59	39	1.512821	4	177	0.022599

##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])

0

1.4. Data Preprocessing

1.4.1 Data Cleaning

1. Subsets of rows and columns with high rates of missing data were removed from the dataset:
   • 4 variables with Fill.Rate<0.9 were excluded for subsequent analysis.
     • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
     • PATRES: Null.Count = 69, Fill.Rate = 0.610
     • ENRTER: Null.Count = 61, Fill.Rate = 0.655
     • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • 14 rows with Missing.Rate>0.2 were exluded for subsequent analysis.
     • COUNTRY=Guadeloupe: Missing.Rate= 0.909
     • COUNTRY=Martinique: Missing.Rate= 0.909
     • COUNTRY=French Guiana: Missing.Rate= 0.909
     • COUNTRY=New Caledonia: Missing.Rate= 0.500
     • COUNTRY=French Polynesia: Missing.Rate= 0.500
     • COUNTRY=Guam: Missing.Rate= 0.500
     • COUNTRY=Puerto Rico: Missing.Rate= 0.409
     • COUNTRY=North Korea: Missing.Rate= 0.227
     • COUNTRY=Somalia: Missing.Rate= 0.227
     • COUNTRY=South Sudan: Missing.Rate= 0.227
     • COUNTRY=Venezuela: Missing.Rate= 0.227
     • COUNTRY=Libya: Missing.Rate= 0.227
     • COUNTRY=Eritrea: Missing.Rate= 0.227
     • COUNTRY=Yemen: Missing.Rate= 0.227
2. No variables were removed due to zero or near-zero variance.
3. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 15/18 predictor (numeric)
       • GDPPER
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/18 predictor (categorical)
       • HDICAT

##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)

Dataset Dimensions: 

(163, 22)

##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)

##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)

Dataset Dimensions: 

(163, 18)

1.4.2 Missing Data Imputation

Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to prior to imputation of the target columns.

Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

1. Missing data for numeric variables were imputed using the iterative imputer algorithm with a linear regression estimator.
   • GDPPER: Null.Count = 1
   • FORARE: Null.Count = 1
   • PM2EXP: Null.Count = 5
2. Missing data for categorical variables were imputed using the most frequent value.
   • HDICAP: Null.Count = 1

##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
                                                  list(cancer_rate_cleaned.dtypes),
                                                  list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
                                                  list(cancer_rate_cleaned.count()),
                                                  list(cancer_rate_cleaned.isna().sum(axis=0))), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count'])
display(cleaned_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count
0	COUNTRY	object	163	163	0
1	CANRAT	category	163	163	0
2	GDPPER	float64	163	162	1
3	URBPOP	float64	163	163	0
4	POPGRO	float64	163	163	0
5	LIFEXP	float64	163	163	0
6	TUBINC	float64	163	163	0
7	DTHCMD	float64	163	163	0
8	AGRLND	float64	163	163	0
9	GHGEMI	float64	163	163	0
10	METEMI	float64	163	163	0
11	FORARE	float64	163	162	1
12	CO2EMI	float64	163	163	0
13	PM2EXP	float64	163	158	5
14	POPDEN	float64	163	163	0
15	GDPCAP	float64	163	163	0
16	HDICAT	category	163	162	1
17	EPISCO	float64	163	163	0

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')

##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')

##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	NaN	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()

##################################
# Defining the parameter of the
# iterative imputer which will estimate 
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
    estimator = lr,
    max_iter = 10,
    tol = 1e-10,
    imputation_order = 'ascending',
    random_state=88888888
)

##################################
# Implementing the iterative imputer 
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)

##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array, 
                                           columns = cancer_rate_cleaned_numeric.columns)

##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	65.867296	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')

##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0], inplace=True)
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)

##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')  

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                                  data_type_list,
                                                  row_count_list,
                                                  non_null_count_list,
                                                  null_count_list,
                                                  fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(imputed_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	GDPPER	float64	163	163	0	1.0
1	URBPOP	float64	163	163	0	1.0
2	POPGRO	float64	163	163	0	1.0
3	LIFEXP	float64	163	163	0	1.0
4	TUBINC	float64	163	163	0	1.0
5	DTHCMD	float64	163	163	0	1.0
6	AGRLND	float64	163	163	0	1.0
7	GHGEMI	float64	163	163	0	1.0
8	METEMI	float64	163	163	0	1.0
9	FORARE	float64	163	163	0	1.0
10	CO2EMI	float64	163	163	0	1.0
11	PM2EXP	float64	163	163	0	1.0
12	POPDEN	float64	163	163	0	1.0
13	GDPCAP	float64	163	163	0	1.0
14	EPISCO	float64	163	163	0	1.0
15	CANRAT	category	163	163	0	1.0
16	HDICAT	category	163	163	0	1.0

1.4.3 Outlier Detection

1. High number of outliers observed for 5 numeric variables with Outlier.Ratio>0.10 and marginal to high Skewness.
   • PM2EXP: Outlier.Count = 37, Outlier.Ratio = 0.226, Skewness=-3.061
   • GHGEMI: Outlier.Count = 27, Outlier.Ratio = 0.165, Skewness=+9.299
   • GDPCAP: Outlier.Count = 22, Outlier.Ratio = 0.134, Skewness=+2.311
   • POPDEN: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+9.972
   • METEMI: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+5.688
2. Minimal number of outliers observed for 5 numeric variables with Outlier.Ratio<0.10 and normal Skewness.
   • TUBINC: Outlier.Count = 12, Outlier.Ratio = 0.073, Skewness=+1.747
   • CO2EMI: Outlier.Count = 11, Outlier.Ratio = 0.067, Skewness=+2.693
   • GDPPER: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+1.554
   • EPISCO: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+0.635
   • CANRAT: Outlier.Count = 2, Outlier.Ratio = 0.012, Skewness=+0.910

##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()

##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1

##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()

##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)

##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                  numeric_skewness_list,
                                                  numeric_outlier_count_list,
                                                  numeric_row_count_list,
                                                  numeric_outlier_ratio_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Skewness',
                                                 'Outlier.Count',
                                                 'Row.Count',
                                                 'Outlier.Ratio'])
display(numeric_column_outlier_summary)

	Numeric.Column.Name	Skewness	Outlier.Count	Row.Count	Outlier.Ratio
0	GDPPER	1.554457	3	163	0.018405
1	URBPOP	-0.212327	0	163	0.000000
2	POPGRO	-0.181666	0	163	0.000000
3	LIFEXP	-0.329704	0	163	0.000000
4	TUBINC	1.747962	12	163	0.073620
5	DTHCMD	0.930709	0	163	0.000000
6	AGRLND	0.035315	0	163	0.000000
7	GHGEMI	9.299960	27	163	0.165644
8	METEMI	5.688689	20	163	0.122699
9	FORARE	0.563015	0	163	0.000000
10	CO2EMI	2.693585	11	163	0.067485
11	PM2EXP	-3.088403	37	163	0.226994
12	POPDEN	9.972806	20	163	0.122699
13	GDPCAP	2.311079	22	163	0.134969
14	EPISCO	0.635994	3	163	0.018405

##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_imputed_numeric, x=column)

1.4.4 Collinearity

Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.

1. Majority of the numeric variables reported moderate to high correlation which were statistically significant.
2. Among pairwise combinations of numeric variables, high Pearson.Correlation.Coefficient values were noted for:
   • GDPPER and GDPCAP: Pearson.Correlation.Coefficient = +0.921
   • GHGEMI and METEMI: Pearson.Correlation.Coefficient = +0.905
3. Among the highly correlated pairs, variables with the lowest correlation against the target variable were removed.
   • GDPPER: Pearson.Correlation.Coefficient = +0.690
   • METEMI: Pearson.Correlation.Coefficient = +0.062
4. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 16 columns (variables)
     • 1/16 metadata (object)
       • COUNTRY
     • 1/16 target (categorical)
       • CANRAT
     • 13/16 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/16 predictor (categorical)
       • HDICAT

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
    f, ax = plt.subplots(figsize=(11, 9))
    sns.heatmap(corr, 
                ax=ax,
                mask=mask,
                annot=True, 
                vmin=-1, 
                vmax=1, 
                center=0,
                cmap='coolwarm', 
                linewidths=1, 
                linecolor='gray', 
                cbar_kws={'orientation': 'horizontal'})  

##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
    cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
        cancer_rate_imputed_numeric.loc[:, numeric_column_a], 
        cancer_rate_imputed_numeric.loc[:, numeric_column_b])

##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))

	Pearson.Correlation.Coefficient	Correlation.PValue
GDPPER_GDPCAP	0.921010	8.158179e-68
GHGEMI_METEMI	0.905121	1.087643e-61
POPGRO_DTHCMD	0.759470	7.124695e-32
GDPPER_LIFEXP	0.755787	2.055178e-31
GDPCAP_EPISCO	0.696707	5.312642e-25
LIFEXP_GDPCAP	0.683834	8.321371e-24
GDPPER_EPISCO	0.680812	1.555304e-23
GDPPER_URBPOP	0.666394	2.781623e-22
GDPPER_CO2EMI	0.654958	2.450029e-21
TUBINC_DTHCMD	0.643615	1.936081e-20
URBPOP_LIFEXP	0.623997	5.669778e-19
LIFEXP_EPISCO	0.620271	1.048393e-18
URBPOP_GDPCAP	0.559181	8.624533e-15
CO2EMI_GDPCAP	0.550221	2.782997e-14
URBPOP_CO2EMI	0.550046	2.846393e-14
LIFEXP_CO2EMI	0.531305	2.951829e-13
URBPOP_EPISCO	0.510131	3.507463e-12
POPGRO_TUBINC	0.442339	3.384403e-09
DTHCMD_PM2EXP	0.283199	2.491837e-04
CO2EMI_EPISCO	0.282734	2.553620e-04

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
    p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
    for col in df.columns:
        for col2 in df.drop(col,axis=1).columns:
            _ , p = stats.pearsonr(df[col],df[col2])
            p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
    return p_matrix

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)                     
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05)) 
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)  

##################################
# Filtering out one among the 
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)

Dataset Dimensions: 

(163, 13)

1.4.5 Shape Transformation

Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.

1. A Yeo-Johnson transformation was applied to all numeric variables to improve distributional shape.
2. Most variables achieved symmetrical distributions with minimal outliers after transformation.
3. One variable which remained skewed even after applying shape transformation was removed.
   • PM2EXP
4. The transformed dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
                                          standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)

##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
                                               columns=cancer_rate_imputed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_transformed_numeric, x=column)

##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)

Dataset Dimensions: 

(163, 12)

1.4.6 Centering and Scaling

1. All numeric variables were transformed using the standardization method to achieve a comparable scale between values.
2. The scaled dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting standardization
# to transform the values of the 
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)

##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
                                          columns=cancer_rate_transformed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_scaled_numeric, x=column)

1.4.7 Data Encoding

1. One-hot encoding was applied to the HDICAP_VH variable resulting to 4 additional columns in the dataset:
   • HDICAP_L
   • HDICAP_M
   • HDICAP_H
   • HDICAP_VH

##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
                                               columns=['HDICAT'])

##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])

1.4.8 Preprocessed Data Description

1. The preprocessed dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 12/18 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 4/18 predictor (categorical)
       • HDICAT_L
       • HDICAT_M
       • HDICAT_H
       • HDICAT_VH

##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')  

##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)

Dataset Dimensions: 

(163, 16)

1.5. Data Exploration

1.5.1 Exploratory Data Analysis

1. Bivariate analysis identified individual predictors with generally positive association to the target variable based on visual inspection.
2. Higher values or higher proportions for the following predictors are associated with the CANRAT HIGH category:
   • URBPOP
   • LIFEXP
   • CO2EMI
   • GDPCAP
   • EPISCO
   • HDICAP_VH=1
3. Decreasing values or smaller proportions for the following predictors are associated with the CANRAT LOW category:
   • POPGRO
   • TUBINC
   • DTHCMD
   • HDICAP_L=0
   • HDICAP_M=0
   • HDICAP_H=0
4. Values for the following predictors are not associated with the CANRAT HIGH or LOW categories:
   • AGRLND
   • GHGEMI
   • FORARE
   • POPDEN

##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = cancer_rate_filtered_row['CANRAT'].to_frame()
cancer_rate_preprocessed_target.reset_index(inplace=True, drop=True)
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[cancer_rate_categorical_encoded.columns]
cancer_rate_preprocessed_categorical_combined = cancer_rate_preprocessed_categorical.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed = cancer_rate_preprocessed.drop(cancer_rate_categorical_encoded.columns, axis=1) 
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.columns
cancer_rate_preprocessed_combined = cancer_rate_preprocessed.join(cancer_rate_preprocessed_target)

##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 6
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 30))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual boxplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
    ax = axes[i]
    ax.boxplot([group[x_variable] for name, group in cancer_rate_preprocessed_combined.groupby(y_variable)])
    ax.set_title(f'{y_variable} Versus {x_variable}')
    ax.set_xlabel(y_variable)
    ax.set_ylabel(x_variable)
    ax.set_xticks(range(1, len(cancer_rate_preprocessed_combined[y_variable].unique()) + 1), ['Low', 'High'])

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

##################################
# Segregating the target
# and predictor variable names
##################################
y_variables = cancer_rate_preprocessed_categorical.columns
x_variable = 'CANRAT'

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 2
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 10))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual stacked column plots
# for all categorical columns
##################################
for i, y_variable in enumerate(y_variables):
    ax = axes[i]
    category_counts = cancer_rate_preprocessed_categorical_combined.groupby([x_variable, y_variable]).size().unstack(fill_value=0)
    category_proportions = category_counts.div(category_counts.sum(axis=1), axis=0)
    category_proportions.plot(kind='bar', stacked=True, ax=ax)
    ax.set_title(f'{x_variable} Versus {y_variable}')
    ax.set_xlabel(x_variable)
    ax.set_ylabel('Proportions')

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

1.5.2 Hypothesis Testing

1. The relationship between the numeric predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: Difference in the means between groups LOW and HIGH is equal to zero
   • Alternative: Difference in the means between groups LOW and HIGH is not equal to zero
2. There is sufficient evidence to conclude of a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable in 9 of the 12 numeric predictors given their high t-test statistic values with reported low p-values less than the significance level of 0.05.
   • GDPCAP: T.Test.Statistic=-11.937, Correlation.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, Correlation.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, Correlation.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, Correlation.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, Correlation.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, Correlation.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, Correlation.PValue=0.000
   • POPGRO: T.Test.Statistic=+4.905, Correlation.PValue=0.000
   • GHGEMI: T.Test.Statistic=-2.243, Correlation.PValue=0.026
3. The relationship between the categorical predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: The categorical predictor is independent of the categorical target variable
   • Alternative: The categorical predictor is dependent of the categorical target variable
4. There is sufficient evidence to conclude of a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable in all 4 categorical predictors given their high chisquare statistic values with reported low p-values less than the significance level of 0.05.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
   • HDICAT_H: ChiSquare.Test.Statistic=13.860, ChiSquare.Test.PValue=0.000
   • HDICAT_M: ChiSquare.Test.Statistic=10.286, ChiSquare.Test.PValue=0.001
   • HDICAT_L: ChiSquare.Test.Statistic=9.081, ChiSquare.Test.PValue=0.002

##################################
# Computing the t-test 
# statistic and p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_ttest_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed_combined
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_predictors
for numeric_column in cancer_rate_preprocessed_numeric_columns:
    group_0 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='Low']
    group_1 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='High']
    cancer_rate_preprocessed_numeric_ttest_target['CANRAT_' + numeric_column] = stats.ttest_ind(
        group_0[numeric_column], 
        group_1[numeric_column], 
        equal_var=True)

##################################
# Formulating the pairwise ttest summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_ttest_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(12))

	T.Test.Statistic	T.Test.PValue
CANRAT_GDPCAP	-11.936988	6.247937e-24
CANRAT_EPISCO	-11.788870	1.605980e-23
CANRAT_LIFEXP	-10.979098	2.754214e-21
CANRAT_TUBINC	9.608760	1.463678e-17
CANRAT_DTHCMD	8.375558	2.552108e-14
CANRAT_CO2EMI	-7.030702	5.537463e-11
CANRAT_URBPOP	-6.541001	7.734940e-10
CANRAT_POPGRO	4.904817	2.269446e-06
CANRAT_GHGEMI	-2.243089	2.625563e-02
CANRAT_FORARE	-1.174143	2.420717e-01
CANRAT_POPDEN	-0.495221	6.211191e-01
CANRAT_AGRLND	-0.047628	9.620720e-01

##################################
# Computing the chisquare
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_chisquare_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed_categorical_combined
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
    contingency_table = pd.crosstab(cancer_rate_preprocessed_categorical[categorical_column], 
                                    cancer_rate_preprocessed_categorical['CANRAT'])
    cancer_rate_preprocessed_categorical_chisquare_target['CANRAT_' + categorical_column] = stats.chi2_contingency(
        contingency_table)[0:2]

##################################
# Formulating the pairwise chisquare summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_chisquare_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['ChiSquare.Test.Statistic', 'ChiSquare.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['ChiSquare.Test.PValue'], ascending=True).head(4))

	ChiSquare.Test.Statistic	ChiSquare.Test.PValue
CANRAT_HDICAT_VH	76.764134	1.926446e-18
CANRAT_HDICAT_M	13.860367	1.969074e-04
CANRAT_HDICAT_L	10.285575	1.340742e-03
CANRAT_HDICAT_H	9.080788	2.583087e-03

1.6. Neural Network Classification Gradient and Weight Updates

1.6.1 Premodelling Data Description

1. All the predictor variables determined to have a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable were used for the subsequent modelling process.
   • GDPCAP: T.Test.Statistic=-11.937, Correlation.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, Correlation.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, Correlation.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, Correlation.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, Correlation.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, Correlation.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, Correlation.PValue=0.000
   • POPGRO: T.Test.Statistic=+4.905, Correlation.PValue=0.000
   • GHGEMI: T.Test.Statistic=-2.243, Correlation.PValue=0.026

##################################
# Assignining all numeric columns
# as model predictors
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_combined
cancer_rate_premodelling.columns

Index(['URBPOP', 'POPGRO', 'LIFEXP', 'TUBINC', 'DTHCMD', 'AGRLND', 'GHGEMI',
       'FORARE', 'CO2EMI', 'POPDEN', 'GDPCAP', 'EPISCO', 'CANRAT'],
      dtype='object')

##################################
# Performing a general exploration of the pre-modelling dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 13)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
FORARE     float64
CO2EMI     float64
POPDEN     float64
GDPCAP     float64
EPISCO     float64
CANRAT    category
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate_premodelling.head()

	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	FORARE	CO2EMI	POPDEN	GDPCAP	EPISCO	CANRAT
0	1.186561	0.075944	1.643195	-1.102296	-0.971464	0.377324	1.388807	-0.467775	1.736841	-2.208974	1.549766	1.306738	High
1	1.207291	0.916022	1.487969	-1.102296	-1.091413	0.043134	0.367038	0.398501	0.943507	-0.989532	1.407752	1.102912	High
2	0.172100	-0.100235	1.537044	-1.275298	-0.836295	1.162279	0.211987	-0.815470	1.031680	-0.007131	1.879374	1.145832	High
3	1.024859	-0.155217	0.664178	-1.696341	-0.903718	0.296508	2.565440	0.259803	1.627748	-0.522844	1.685426	0.739753	High
4	1.271466	-0.718131	1.381877	-1.413414	-0.657145	1.162434	0.019979	-0.559264	0.686270	0.512619	1.657777	2.218327	High

##################################
# Formulating a scatterplot matrix
# of all pairwise combinations of 
# numeric predictors labeled by
# categorical response classes
##################################
sns.pairplot(cancer_rate_premodelling, hue='CANRAT')
plt.show()

##################################
# Converting the dataframe to
# a numpy array
##################################
cancer_rate_premodelling_matrix = cancer_rate_premodelling.to_numpy()

##################################
# Preparing the data and
# and converting to a suitable format
# as a neural network model input
##################################
matrix_x_values = cancer_rate_premodelling.iloc[:,0:12].to_numpy()
y_values_series = np.where(cancer_rate_premodelling['CANRAT'] == 'High', 1, 0)
y_values = pd.get_dummies(y_values_series)
y_values = y_values.to_numpy()

1.6.2 No Regularization

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Regularization Algorithms, in the context of neural network classification, are techniques used to prevent overfitting and improve the generalization performance of the model by imposing constraints on its parameters during training. These constraints are typically applied to the weights of the neural network and are aimed at reducing model complexity, controlling the magnitude of the weights, and promoting simpler and more generalizable solutions. Regularization approaches work by adding penalty terms to the loss function during training. These penalty terms penalize large weights or complex models, encouraging the optimization process to prioritize simpler solutions that generalize well to unseen data. By doing so, regularization helps prevent the neural network from fitting noise or irrelevant patterns present in the training data and encourages it to learn more robust and meaningful representations.

No Regularization provides no additional constraints or penalties imposed on the model's parameters (weights and biases) during training. The model is free to learn complex patterns in the training data without any restrictions. Without regularization, the neural network may have a tendency to memorize noise or irrelevant patterns present in the training data. This can result in a highly complex model with large weights and intricate decision boundaries. Due to the lack of constraints on model complexity, there is a higher risk of overfitting when the model captures noise or idiosyncrasies in the training set rather than learning the underlying patterns of the data. In effect, the model may perform well on the training data but poorly on new, unseen data.

##################################
# Defining the neural network architecture
##################################
input_size = 12
hidden_sizes = [3, 3, 3]
output_size = 2

##################################
# Defining the training parameters
##################################
learning_rate = 0.01
iterations = 5001

##################################
# Initializing model weights and biases
##################################
def initialize_parameters(input_size, hidden_sizes, output_size):
    parameters = {}
    layer_sizes = [input_size] + hidden_sizes + [output_size]
    for i in range(1, len(layer_sizes)):
        parameters[f'W{i}'] = np.random.randn(layer_sizes[i-1], layer_sizes[i])
        parameters[f'b{i}'] = np.zeros((1, layer_sizes[i]))
    return parameters

##################################
# Defining the activation function (ReLU)
##################################
def relu(x):
    return np.maximum(0, x)

##################################
# Defining the activation function (Sigmoid)
##################################
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

##################################
# Defining the Softmax function
##################################
def softmax(x):
    exps = np.exp(x - np.max(x, axis=1, keepdims=True))
    return exps / np.sum(exps, axis=1, keepdims=True)

##################################
# Defining the Forward propagation algorithm
##################################
def forward_propagation(X, parameters):
    cache = {'A0': X}
    for i in range(1, len(parameters) // 2 + 1):
        cache[f'Z{i}'] = np.dot(cache[f'A{i-1}'], parameters[f'W{i}']) + parameters[f'b{i}']
        cache[f'A{i}'] = relu(cache[f'Z{i}']) if i != len(parameters) // 2 else softmax(cache[f'Z{i}'])
    return cache

##################################
# Defining the Backward propagation algorithm
##################################
def backward_propagation(X, Y, parameters, cache):
    m = X.shape[0]
    gradients = {}
    dZ = cache[f'A{len(parameters) // 2}'] - Y
    for i in range(len(parameters) // 2, 0, -1):
        gradients[f'dW{i}'] = (1 / m) * np.dot(cache[f'A{i-1}'].T, dZ)
        gradients[f'db{i}'] = (1 / m) * np.sum(dZ, axis=0, keepdims=True)
        if i > 1:
            dA = np.dot(dZ, parameters[f'W{i}'].T)
            dZ = dA * (cache[f'Z{i-1}'] > 0)
    return gradients

##################################
# Defining the Cross-entropy loss
##################################
def compute_cost(Y, Y_hat):
    m = Y.shape[0]
    logprobs = -np.log(Y_hat[range(m), np.argmax(Y, axis=1)])
    return np.sum(logprobs) / m

##################################
# Updating model parameters
# with no regularization
##################################
def update_parameters_no_reg(parameters, gradients, learning_rate):
    for i in range(1, len(parameters) // 2 + 1):
        parameters[f'W{i}'] -= learning_rate * gradients[f'dW{i}']
        parameters[f'b{i}'] -= learning_rate * gradients[f'db{i}']
    return parameters

##################################
# Implementing neural network model training
# using no regularization
##################################

##################################
# Initializing training parameters
##################################
np.random.seed(88888)
parameters = initialize_parameters(input_size, hidden_sizes, output_size)

##################################
# Creating lists to store cost and accuracy for plotting
##################################
costs = []
accuracies = []

##################################
# Creating lists to store weights for plotting
##################################
weight_history = {f'W{i}': [] for i in range(1, len(hidden_sizes) + 2)}

##################################
# Training a neural network model
# using no regularization
##################################
for i in range(iterations):
    # Implementing forward propagation
    cache = forward_propagation(matrix_x_values, parameters)
    Y_hat = cache[f'A{len(parameters) // 2}']
    
    # Computing cost and accuracy
    cost = compute_cost(y_values, Y_hat)
    accuracy = np.mean(np.argmax(y_values, axis=1) == np.argmax(Y_hat, axis=1))
    
    # Implementing backward propagation
    gradients = backward_propagation(matrix_x_values, y_values, parameters, cache)
    
    # Updating model parameter values
    parameters = update_parameters_no_reg(parameters, gradients, learning_rate)
    
    # Recording model weight values
    for j in range(1, len(hidden_sizes) + 2):
        weight_history[f'W{j}'].append(parameters[f'W{j}'].copy())
    
    # Recording cost and accuracy values
    costs.append(cost)
    accuracies.append(accuracy)
    
    # Printing cost and accuracy every 100 iterations
    if i % 100 == 0:
        print(f"Iteration {i}: Cost = {cost}, Accuracy = {accuracy}")

Iteration 0: Cost = 0.6655730513519705, Accuracy = 0.7484662576687117
Iteration 100: Cost = 0.44337999165463543, Accuracy = 0.7484662576687117
Iteration 200: Cost = 0.4296527297741833, Accuracy = 0.7484662576687117
Iteration 300: Cost = 0.42070497080190816, Accuracy = 0.7484662576687117
Iteration 400: Cost = 0.413061282503375, Accuracy = 0.7484662576687117
Iteration 500: Cost = 0.4061728704817939, Accuracy = 0.7484662576687117
Iteration 600: Cost = 0.39995724221500645, Accuracy = 0.7484662576687117
Iteration 700: Cost = 0.39458769409625294, Accuracy = 0.7484662576687117
Iteration 800: Cost = 0.38949596166840167, Accuracy = 0.7484662576687117
Iteration 900: Cost = 0.38497517328300895, Accuracy = 0.7484662576687117
Iteration 1000: Cost = 0.3809436200108279, Accuracy = 0.7975460122699386
Iteration 1100: Cost = 0.37727075177667163, Accuracy = 0.7852760736196319
Iteration 1200: Cost = 0.37387392095786326, Accuracy = 0.7914110429447853
Iteration 1300: Cost = 0.3706503011149609, Accuracy = 0.7791411042944786
Iteration 1400: Cost = 0.36756963290365535, Accuracy = 0.7852760736196319
Iteration 1500: Cost = 0.36458404147756973, Accuracy = 0.803680981595092
Iteration 1600: Cost = 0.3612766252841342, Accuracy = 0.8466257668711656
Iteration 1700: Cost = 0.357134360433017, Accuracy = 0.852760736196319
Iteration 1800: Cost = 0.35269802361655544, Accuracy = 0.8711656441717791
Iteration 1900: Cost = 0.34846303622133556, Accuracy = 0.8711656441717791
Iteration 2000: Cost = 0.3436917696319817, Accuracy = 0.8711656441717791
Iteration 2100: Cost = 0.33842245819855354, Accuracy = 0.8711656441717791
Iteration 2200: Cost = 0.33296683643984326, Accuracy = 0.8711656441717791
Iteration 2300: Cost = 0.32707243317904855, Accuracy = 0.8773006134969326
Iteration 2400: Cost = 0.3202156716142601, Accuracy = 0.8711656441717791
Iteration 2500: Cost = 0.31345579274924, Accuracy = 0.8711656441717791
Iteration 2600: Cost = 0.3069538408163979, Accuracy = 0.8711656441717791
Iteration 2700: Cost = 0.3008522153030943, Accuracy = 0.8834355828220859
Iteration 2800: Cost = 0.29504807562670776, Accuracy = 0.8773006134969326
Iteration 2900: Cost = 0.28949665746105524, Accuracy = 0.8834355828220859
Iteration 3000: Cost = 0.2834331137306993, Accuracy = 0.8834355828220859
Iteration 3100: Cost = 0.2773402269258411, Accuracy = 0.8895705521472392
Iteration 3200: Cost = 0.27144698132147027, Accuracy = 0.8895705521472392
Iteration 3300: Cost = 0.26569904906368097, Accuracy = 0.8895705521472392
Iteration 3400: Cost = 0.2601293462373859, Accuracy = 0.8895705521472392
Iteration 3500: Cost = 0.25519554148903284, Accuracy = 0.8895705521472392
Iteration 3600: Cost = 0.2504756963819801, Accuracy = 0.8895705521472392
Iteration 3700: Cost = 0.24553061546407004, Accuracy = 0.8957055214723927
Iteration 3800: Cost = 0.24082289556588482, Accuracy = 0.8957055214723927
Iteration 3900: Cost = 0.23648196660441367, Accuracy = 0.8957055214723927
Iteration 4000: Cost = 0.2325581819214636, Accuracy = 0.8957055214723927
Iteration 4100: Cost = 0.22900452092328547, Accuracy = 0.901840490797546
Iteration 4200: Cost = 0.22554933649437556, Accuracy = 0.901840490797546
Iteration 4300: Cost = 0.22237283262286484, Accuracy = 0.901840490797546
Iteration 4400: Cost = 0.21956579583200125, Accuracy = 0.901840490797546
Iteration 4500: Cost = 0.21687063564756745, Accuracy = 0.9079754601226994
Iteration 4600: Cost = 0.21428226254644192, Accuracy = 0.9079754601226994
Iteration 4700: Cost = 0.21185730328243121, Accuracy = 0.9141104294478528
Iteration 4800: Cost = 0.20951426452485125, Accuracy = 0.9202453987730062
Iteration 4900: Cost = 0.2072391302349501, Accuracy = 0.9202453987730062
Iteration 5000: Cost = 0.20503671639364385, Accuracy = 0.9202453987730062

##################################
# Plotting cost and accuracy profiles
##################################
plt.figure(figsize=(15, 4.75))
plt.subplot(1, 2, 1)
plt.plot(range(iterations), costs)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.title('No Regularization: Cost Function by Iteration')

plt.subplot(1, 2, 2)
plt.plot(range(iterations), accuracies)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.title('No Regularization: Classification Accuracy by Iteration')

plt.show()

##################################
# Plotting model weight profiles
##################################
num_layers = len(hidden_sizes) + 1
plt.figure(figsize=(15, 10))
for i in range(1, num_layers + 1):
    plt.subplot(2, num_layers // 2, i)
    plt.plot(range(iterations), [np.max(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Maximum Weight")
    plt.plot(range(iterations), [np.mean(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Mean Weight")
    plt.plot(range(iterations), [np.min(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Minimum Weight")
    plt.ylim([-1,5])
    plt.legend(loc="upper right")
    plt.xlabel('Iterations')
    plt.ylabel('Average Absolute Weight')
    plt.title(f'No Regularization: Layer {i} Weights by Iteration')

plt.show()

##################################
# Gathering the final accuracy, cost 
# and mean layer weight values for 
# No Regularization
##################################
NR_metrics = pd.DataFrame(["ACCURACY","LOSS","LAYER 1 MEAN WEIGHT","LAYER 2 MEAN WEIGHT","LAYER 3 MEAN WEIGHT","LAYER 4 MEAN WEIGHT"])
NR_values = pd.DataFrame([accuracies[-1],
                          costs[-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W1']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W2']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W3']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W4']][-1]])
NR_method = pd.DataFrame(["No Regularization"]*6)
NR_summary = pd.concat([NR_metrics, 
                        NR_values,
                        NR_method], axis=1)
NR_summary.columns = ['Metric', 'Value', 'Method']
NR_summary.reset_index(inplace=True, drop=True)
display(NR_summary)

	Metric	Value	Method
0	ACCURACY	0.920245	No Regularization
1	LOSS	0.205037	No Regularization
2	LAYER 1 MEAN WEIGHT	0.878401	No Regularization
3	LAYER 2 MEAN WEIGHT	0.663882	No Regularization
4	LAYER 3 MEAN WEIGHT	0.662323	No Regularization
5	LAYER 4 MEAN WEIGHT	1.042851	No Regularization

1.6.3 L1 Regularization

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Regularization Algorithms, in the context of neural network classification, are techniques used to prevent overfitting and improve the generalization performance of the model by imposing constraints on its parameters during training. These constraints are typically applied to the weights of the neural network and are aimed at reducing model complexity, controlling the magnitude of the weights, and promoting simpler and more generalizable solutions. Regularization approaches work by adding penalty terms to the loss function during training. These penalty terms penalize large weights or complex models, encouraging the optimization process to prioritize simpler solutions that generalize well to unseen data. By doing so, regularization helps prevent the neural network from fitting noise or irrelevant patterns present in the training data and encourages it to learn more robust and meaningful representations.

L1 Regularization adds a penalty term to the loss function proportional to the absolute values of the weights. It encourages sparsity in the weight matrix, effectively performing feature selection by driving some weights to exactly zero. This makes the process robust to irrelevant features by reducing their impact on the model. Despite these advantages, L1 regularization has drawbacks as a method including less stable solutions (the optimization problem with L1 regularization can have multiple solutions, leading to instability in the learned weights) and being not differentiable (the absolute value function used in L1 regularization is not differentiable at zero, which can complicate optimization).

##################################
# Updating model parameters
# with L1 regularization
##################################
def update_parameters_l1(parameters, gradients, learning_rate, lambd):
    for i in range(1, len(parameters) // 2 + 1):
        parameters[f'W{i}'] -= learning_rate * (gradients[f'dW{i}'] + lambd * np.sign(parameters[f'W{i}']))
        parameters[f'b{i}'] -= learning_rate * gradients[f'db{i}']
    return parameters

##################################
# Implementing neural network model training
# using L1 regularization
##################################

##################################
# Initializing training parameters
##################################
np.random.seed(88888)
parameters = initialize_parameters(input_size, hidden_sizes, output_size)

##################################
# Initializing regularization parameters
##################################
lambd = 0.01

##################################
# Creating lists to store cost and accuracy for plotting
##################################
costs = []
accuracies = []

##################################
# Creating lists to store weights for plotting
##################################
weight_history = {f'W{i}': [] for i in range(1, len(hidden_sizes) + 2)}

##################################
# Training a neural network model
# using L1 regularization
##################################
for i in range(iterations):
    # Implementing forward propagation
    cache = forward_propagation(matrix_x_values, parameters)
    Y_hat = cache[f'A{len(parameters) // 2}']
    
    # Computing cost and accuracy
    cost = compute_cost(y_values, Y_hat)
    accuracy = np.mean(np.argmax(y_values, axis=1) == np.argmax(Y_hat, axis=1))
    
    # Implementing backward propagation
    gradients = backward_propagation(matrix_x_values, y_values, parameters, cache)
    
    # Updating model parameter values
    parameters = update_parameters_l1(parameters, gradients, learning_rate, lambd)
    
    # Recording model weight values
    for j in range(1, len(hidden_sizes) + 2):
        weight_history[f'W{j}'].append(parameters[f'W{j}'].copy())
    
    # Recording cost and accuracy values
    costs.append(cost)
    accuracies.append(accuracy)
    
    # Printing cost and accuracy every 100 iterations
    if i % 100 == 0:
        print(f"Iteration {i}: Cost = {cost}, Accuracy = {accuracy}")

Iteration 0: Cost = 0.6655730513519705, Accuracy = 0.7484662576687117
Iteration 100: Cost = 0.4462035799912659, Accuracy = 0.7484662576687117
Iteration 200: Cost = 0.43307112414387683, Accuracy = 0.7484662576687117
Iteration 300: Cost = 0.42457535971007343, Accuracy = 0.7484662576687117
Iteration 400: Cost = 0.41765452103112666, Accuracy = 0.7484662576687117
Iteration 500: Cost = 0.41170724984703505, Accuracy = 0.7484662576687117
Iteration 600: Cost = 0.4063499603006234, Accuracy = 0.7484662576687117
Iteration 700: Cost = 0.40141996484045106, Accuracy = 0.7484662576687117
Iteration 800: Cost = 0.3969570506741036, Accuracy = 0.7484662576687117
Iteration 900: Cost = 0.39285529655789025, Accuracy = 0.7484662576687117
Iteration 1000: Cost = 0.38913393420566117, Accuracy = 0.7484662576687117
Iteration 1100: Cost = 0.38571148027548785, Accuracy = 0.7484662576687117
Iteration 1200: Cost = 0.3825210045646434, Accuracy = 0.7914110429447853
Iteration 1300: Cost = 0.37937159603685555, Accuracy = 0.803680981595092
Iteration 1400: Cost = 0.37642648361234926, Accuracy = 0.803680981595092
Iteration 1500: Cost = 0.37341353354514417, Accuracy = 0.803680981595092
Iteration 1600: Cost = 0.3704330888484434, Accuracy = 0.8098159509202454
Iteration 1700: Cost = 0.3677665430279169, Accuracy = 0.8098159509202454
Iteration 1800: Cost = 0.36536349299758103, Accuracy = 0.8159509202453987
Iteration 1900: Cost = 0.3631082869687799, Accuracy = 0.803680981595092
Iteration 2000: Cost = 0.360950443909029, Accuracy = 0.8159509202453987
Iteration 2100: Cost = 0.3588633922570358, Accuracy = 0.8466257668711656
Iteration 2200: Cost = 0.3566944719925671, Accuracy = 0.8588957055214724
Iteration 2300: Cost = 0.3543498283485834, Accuracy = 0.8650306748466258
Iteration 2400: Cost = 0.35192154110725726, Accuracy = 0.8773006134969326
Iteration 2500: Cost = 0.3495143995277487, Accuracy = 0.8834355828220859
Iteration 2600: Cost = 0.34706384497444936, Accuracy = 0.8834355828220859
Iteration 2700: Cost = 0.34402578656122357, Accuracy = 0.8834355828220859
Iteration 2800: Cost = 0.3404546577239142, Accuracy = 0.8895705521472392
Iteration 2900: Cost = 0.33633340551158214, Accuracy = 0.8895705521472392
Iteration 3000: Cost = 0.3317444198337502, Accuracy = 0.8895705521472392
Iteration 3100: Cost = 0.3264606040483411, Accuracy = 0.8957055214723927
Iteration 3200: Cost = 0.32077768315962374, Accuracy = 0.901840490797546
Iteration 3300: Cost = 0.31478413087215135, Accuracy = 0.901840490797546
Iteration 3400: Cost = 0.3078686074258959, Accuracy = 0.901840490797546
Iteration 3500: Cost = 0.30064015748204737, Accuracy = 0.901840490797546
Iteration 3600: Cost = 0.29328995535853325, Accuracy = 0.9079754601226994
Iteration 3700: Cost = 0.28584435234367245, Accuracy = 0.9079754601226994
Iteration 3800: Cost = 0.27821774403717314, Accuracy = 0.9079754601226994
Iteration 3900: Cost = 0.26988387941259606, Accuracy = 0.9141104294478528
Iteration 4000: Cost = 0.26115814864046644, Accuracy = 0.9141104294478528
Iteration 4100: Cost = 0.2528971596046284, Accuracy = 0.9141104294478528
Iteration 4200: Cost = 0.24518856179189225, Accuracy = 0.9079754601226994
Iteration 4300: Cost = 0.2370974152519641, Accuracy = 0.9079754601226994
Iteration 4400: Cost = 0.22830112643766665, Accuracy = 0.9079754601226994
Iteration 4500: Cost = 0.2192976829867574, Accuracy = 0.9079754601226994
Iteration 4600: Cost = 0.21094110711087927, Accuracy = 0.9141104294478528
Iteration 4700: Cost = 0.20290464572348446, Accuracy = 0.9202453987730062
Iteration 4800: Cost = 0.19480835010431513, Accuracy = 0.9263803680981595
Iteration 4900: Cost = 0.1871322306145051, Accuracy = 0.9386503067484663
Iteration 5000: Cost = 0.17995578256489161, Accuracy = 0.9447852760736196

##################################
# Plotting cost and accuracy profiles
##################################
plt.figure(figsize=(15, 4.75))
plt.subplot(1, 2, 1)
plt.plot(range(iterations), costs)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.title('L1 Regularization: Cost Function by Iteration')

plt.subplot(1, 2, 2)
plt.plot(range(iterations), accuracies)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.title('L1 Regularization: Classification Accuracy by Iteration')

plt.show()

##################################
# Plotting model weight profiles
##################################
num_layers = len(hidden_sizes) + 1
plt.figure(figsize=(15, 10))
for i in range(1, num_layers + 1):
    plt.subplot(2, num_layers // 2, i)
    plt.plot(range(iterations), [np.max(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Maximum Weight")
    plt.plot(range(iterations), [np.mean(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Mean Weight")
    plt.plot(range(iterations), [np.min(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Minimum Weight")
    plt.ylim([-1,5])
    plt.legend(loc="upper right")
    plt.xlabel('Iterations')
    plt.ylabel('Average Absolute Weight')
    plt.title(f'L1 Regularization: Layer {i} Weights by Iteration')

plt.show()

##################################
# Gathering the final accuracy, cost 
# and mean layer weight values for 
# L1 Regularization
##################################
L1R_metrics = pd.DataFrame(["ACCURACY","LOSS","LAYER 1 MEAN WEIGHT","LAYER 2 MEAN WEIGHT","LAYER 3 MEAN WEIGHT","LAYER 4 MEAN WEIGHT"])
L1R_values = pd.DataFrame([accuracies[-1],
                          costs[-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W1']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W2']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W3']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W4']][-1]])
L1R_method = pd.DataFrame(["L1 Regularization"]*6)
L1R_summary = pd.concat([L1R_metrics, 
                         L1R_values,
                         L1R_method], axis=1)
L1R_summary.columns = ['Metric', 'Value', 'Method']
L1R_summary.reset_index(inplace=True, drop=True)
display(L1R_summary)

	Metric	Value	Method
0	ACCURACY	0.944785	L1 Regularization
1	LOSS	0.179956	L1 Regularization
2	LAYER 1 MEAN WEIGHT	0.493354	L1 Regularization
3	LAYER 2 MEAN WEIGHT	0.413750	L1 Regularization
4	LAYER 3 MEAN WEIGHT	0.390737	L1 Regularization
5	LAYER 4 MEAN WEIGHT	0.715698	L1 Regularization

1.6.4 L2 Regularization

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Regularization Algorithms, in the context of neural network classification, are techniques used to prevent overfitting and improve the generalization performance of the model by imposing constraints on its parameters during training. These constraints are typically applied to the weights of the neural network and are aimed at reducing model complexity, controlling the magnitude of the weights, and promoting simpler and more generalizable solutions. Regularization approaches work by adding penalty terms to the loss function during training. These penalty terms penalize large weights or complex models, encouraging the optimization process to prioritize simpler solutions that generalize well to unseen data. By doing so, regularization helps prevent the neural network from fitting noise or irrelevant patterns present in the training data and encourages it to learn more robust and meaningful representations.

L2 Regularization adds a penalty term to the loss function proportional to the square of the weights. It discourages large weights and pushes them towards smaller values. The process provides stable solutions as L2 regularization leads to a convex optimization problem, ensuring a unique and stable solution. Additionally, the method has a smoothing effect by encourages small weights, which can prevent overfitting and lead to smoother decision boundaries. L2 regularization however does not perform feature selection by not forcing any weights to become exactly zero, so it may not eliminate irrelevant features. As a method, it also may not effectively handle highly correlated features well since L2 regularization treats all features equally.

##################################
# Updating model parameters
# with L2 regularization
##################################
def update_parameters_l2(parameters, gradients, learning_rate, lambd):
    for i in range(1, len(parameters) // 2 + 1):
        parameters[f'W{i}'] -= learning_rate * (gradients[f'dW{i}'] + lambd * parameters[f'W{i}'])
        parameters[f'b{i}'] -= learning_rate * gradients[f'db{i}']
    return parameters

##################################
# Implementing neural network model training
# using L2 regularization
##################################

##################################
# Initializing training parameters
##################################
np.random.seed(88888)
parameters = initialize_parameters(input_size, hidden_sizes, output_size)

##################################
# Initializing regularization parameters
##################################
lambd = 0.01

##################################
# Creating lists to store cost and accuracy for plotting
##################################
costs = []
accuracies = []

##################################
# Creating lists to store weights for plotting
##################################
weight_history = {f'W{i}': [] for i in range(1, len(hidden_sizes) + 2)}

##################################
# Training a neural network model
# using L2 regularization
##################################
for i in range(iterations):
    # Implementing forward propagation
    cache = forward_propagation(matrix_x_values, parameters)
    Y_hat = cache[f'A{len(parameters) // 2}']
    
    # Computing cost and accuracy
    cost = compute_cost(y_values, Y_hat)
    accuracy = np.mean(np.argmax(y_values, axis=1) == np.argmax(Y_hat, axis=1))
    
    # Implementing backward propagation
    gradients = backward_propagation(matrix_x_values, y_values, parameters, cache)
    
    # Updating model parameter values
    parameters = update_parameters_l2(parameters, gradients, learning_rate, lambd)
    
    # Recording model weight values
    for j in range(1, len(hidden_sizes) + 2):
        weight_history[f'W{j}'].append(parameters[f'W{j}'].copy())
    
    # Recording cost and accuracy values
    costs.append(cost)
    accuracies.append(accuracy)
    
    # Printing cost and accuracy every 100 iterations
    if i % 100 == 0:
        print(f"Iteration {i}: Cost = {cost}, Accuracy = {accuracy}")

Iteration 0: Cost = 0.6655730513519705, Accuracy = 0.7484662576687117
Iteration 100: Cost = 0.44429964390155074, Accuracy = 0.7484662576687117
Iteration 200: Cost = 0.43045876994697085, Accuracy = 0.7484662576687117
Iteration 300: Cost = 0.4216253632457659, Accuracy = 0.7484662576687117
Iteration 400: Cost = 0.41429556819902924, Accuracy = 0.7484662576687117
Iteration 500: Cost = 0.407808510163234, Accuracy = 0.7484662576687117
Iteration 600: Cost = 0.4019708537957727, Accuracy = 0.7484662576687117
Iteration 700: Cost = 0.39684543072268164, Accuracy = 0.7484662576687117
Iteration 800: Cost = 0.3920371113720447, Accuracy = 0.7484662576687117
Iteration 900: Cost = 0.38778015562298945, Accuracy = 0.7484662576687117
Iteration 1000: Cost = 0.38395598783499835, Accuracy = 0.7484662576687117
Iteration 1100: Cost = 0.38044959933609496, Accuracy = 0.7975460122699386
Iteration 1200: Cost = 0.3771724996008687, Accuracy = 0.7914110429447853
Iteration 1300: Cost = 0.37407878873036193, Accuracy = 0.7852760736196319
Iteration 1400: Cost = 0.3711554982481661, Accuracy = 0.7975460122699386
Iteration 1500: Cost = 0.3683436832278206, Accuracy = 0.7791411042944786
Iteration 1600: Cost = 0.3655033686680177, Accuracy = 0.7791411042944786
Iteration 1700: Cost = 0.36275189306899513, Accuracy = 0.7852760736196319
Iteration 1800: Cost = 0.3597398271746222, Accuracy = 0.8098159509202454
Iteration 1900: Cost = 0.3566011955167585, Accuracy = 0.8404907975460123
Iteration 2000: Cost = 0.35288720939630647, Accuracy = 0.852760736196319
Iteration 2100: Cost = 0.3484851330217932, Accuracy = 0.8711656441717791
Iteration 2200: Cost = 0.3432958874841974, Accuracy = 0.8711656441717791
Iteration 2300: Cost = 0.33760321157554923, Accuracy = 0.8711656441717791
Iteration 2400: Cost = 0.33180949325055176, Accuracy = 0.8834355828220859
Iteration 2500: Cost = 0.3261602386561661, Accuracy = 0.8834355828220859
Iteration 2600: Cost = 0.3204014679517702, Accuracy = 0.8834355828220859
Iteration 2700: Cost = 0.3144087316835841, Accuracy = 0.8895705521472392
Iteration 2800: Cost = 0.3082058865342837, Accuracy = 0.8895705521472392
Iteration 2900: Cost = 0.301591694439833, Accuracy = 0.8895705521472392
Iteration 3000: Cost = 0.29446005073465753, Accuracy = 0.901840490797546
Iteration 3100: Cost = 0.2875654463067539, Accuracy = 0.901840490797546
Iteration 3200: Cost = 0.2807765512862666, Accuracy = 0.901840490797546
Iteration 3300: Cost = 0.27309660385270446, Accuracy = 0.901840490797546
Iteration 3400: Cost = 0.26454810911961407, Accuracy = 0.9079754601226994
Iteration 3500: Cost = 0.25538936911769283, Accuracy = 0.9141104294478528
Iteration 3600: Cost = 0.2470696004687147, Accuracy = 0.9141104294478528
Iteration 3700: Cost = 0.23946139524956278, Accuracy = 0.9079754601226994
Iteration 3800: Cost = 0.23268259797072838, Accuracy = 0.9079754601226994
Iteration 3900: Cost = 0.22675946664763505, Accuracy = 0.9079754601226994
Iteration 4000: Cost = 0.22160929548992525, Accuracy = 0.9141104294478528
Iteration 4100: Cost = 0.2167888900502071, Accuracy = 0.9141104294478528
Iteration 4200: Cost = 0.21222808925582234, Accuracy = 0.9141104294478528
Iteration 4300: Cost = 0.20792279041601808, Accuracy = 0.9141104294478528
Iteration 4400: Cost = 0.20393108475817381, Accuracy = 0.9202453987730062
Iteration 4500: Cost = 0.20016016310514442, Accuracy = 0.9263803680981595
Iteration 4600: Cost = 0.19424766890014947, Accuracy = 0.9202453987730062
Iteration 4700: Cost = 0.1883268128700009, Accuracy = 0.9325153374233128
Iteration 4800: Cost = 0.18382694569588456, Accuracy = 0.9325153374233128
Iteration 4900: Cost = 0.1790868990533559, Accuracy = 0.9447852760736196
Iteration 5000: Cost = 0.17509776446937939, Accuracy = 0.9447852760736196

##################################
# Plotting cost and accuracy profiles
##################################
plt.figure(figsize=(15, 4.75))
plt.subplot(1, 2, 1)
plt.plot(range(iterations), costs)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.title('L2 Regularization: Cost Function by Iteration')

plt.subplot(1, 2, 2)
plt.plot(range(iterations), accuracies)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.title('L2 Regularization: Classification Accuracy by Iteration')

plt.show()

##################################
# Plotting model weight profiles
##################################
num_layers = len(hidden_sizes) + 1
plt.figure(figsize=(15, 10))
for i in range(1, num_layers + 1):
    plt.subplot(2, num_layers // 2, i)
    plt.plot(range(iterations), [np.max(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Maximum Weight")
    plt.plot(range(iterations), [np.mean(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Mean Weight")
    plt.plot(range(iterations), [np.min(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Minimum Weight")
    plt.ylim([-1,5])
    plt.legend(loc="upper right")
    plt.xlabel('Iterations')
    plt.ylabel('Average Absolute Weight')
    plt.title(f'L2 Regularization: Layer {i} Weights by Iteration')

plt.show()

##################################
# Gathering the final accuracy, cost 
# and mean layer weight values for 
# L2 Regularization
##################################
L2R_metrics = pd.DataFrame(["ACCURACY","LOSS","LAYER 1 MEAN WEIGHT","LAYER 2 MEAN WEIGHT","LAYER 3 MEAN WEIGHT","LAYER 4 MEAN WEIGHT"])
L2R_values = pd.DataFrame([accuracies[-1],
                          costs[-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W1']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W2']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W3']][-1],
                          [np.mean(np.abs(weights)) for weights in weight_history['W4']][-1]])
L2R_method = pd.DataFrame(["L1 Regularization"]*6)
L2R_summary = pd.concat([L2R_metrics, 
                         L2R_values,
                         L2R_method], axis=1)
L2R_summary.columns = ['Metric', 'Value', 'Method']
L2R_summary.reset_index(inplace=True, drop=True)
display(L2R_summary)

	Metric	Value	Method
0	ACCURACY	0.944785	L1 Regularization
1	LOSS	0.175098	L1 Regularization
2	LAYER 1 MEAN WEIGHT	0.537587	L1 Regularization
3	LAYER 2 MEAN WEIGHT	0.503285	L1 Regularization
4	LAYER 3 MEAN WEIGHT	0.487109	L1 Regularization
5	LAYER 4 MEAN WEIGHT	0.749369	L1 Regularization

1.6.5 ElasticNet Regularization

Backpropagation and Weight Update, in the context of an artificial neural network, involve the process of iteratively adjusting the weights of the connections between neurons in the network to minimize the difference between the predicted and the actual target responses. Input data is fed into the neural network, and it propagates through the network layer by layer, starting from the input layer, through hidden layers, and ending at the output layer. At each neuron, the weighted sum of inputs is calculated, followed by the application of an activation function to produce the neuron's output. Once the forward pass is complete, the network's output is compared to the actual target output. The difference between the predicted output and the actual output is quantified using a loss function, which measures the discrepancy between the predicted and actual values. Common loss functions for classification tasks include cross-entropy loss. During the backward pass, the error is propagated backward through the network to compute the gradients of the loss function with respect to each weight in the network. This is achieved using the chain rule of calculus, which allows the error to be decomposed and distributed backward through the network. The gradients quantify how much a change in each weight would affect the overall error of the network. Once the gradients are computed, the weights are updated in the opposite direction of the gradient to minimize the error. This update is typically performed using an optimization algorithm such as gradient descent, which adjusts the weights in proportion to their gradients and a learning rate hyperparameter. The learning rate determines the size of the step taken in the direction opposite to the gradient. These steps are repeated for multiple iterations (epochs) over the training data. As the training progresses, the weights are adjusted iteratively to minimize the error, leading to a neural network model that accurately classifies input data.

Regularization Algorithms, in the context of neural network classification, are techniques used to prevent overfitting and improve the generalization performance of the model by imposing constraints on its parameters during training. These constraints are typically applied to the weights of the neural network and are aimed at reducing model complexity, controlling the magnitude of the weights, and promoting simpler and more generalizable solutions. Regularization approaches work by adding penalty terms to the loss function during training. These penalty terms penalize large weights or complex models, encouraging the optimization process to prioritize simpler solutions that generalize well to unseen data. By doing so, regularization helps prevent the neural network from fitting noise or irrelevant patterns present in the training data and encourages it to learn more robust and meaningful representations.

ElasticNet Regularization combines L1 and L2 regularization by adding both penalty terms to the loss function. It addresses the limitations of L1 and L2 regularization by providing a compromise between feature selection and weight shrinkage. ElasticNet regularization combines the advantages of both L1 and L2 methods by providing a more flexible regularization approach. It can handle highly correlated features better than L1 regularization alone. Some disadvantages include the presence of more hyperparameters (ElasticNet regularization introduces an additional hyperparameter to control the balance between L1 and L2 penalties, which needs to be tuned) and computational complexity (training with ElasticNet regularization can be computationally more expensive compared to L1 or L2 regularization alone).

##################################
# Updating model parameters
# with ElasticNet regularization
##################################
def update_parameters_elastic(parameters, gradients, learning_rate, lambd):
    for i in range(1, len(parameters) // 2 + 1):
        parameters[f'W{i}'] -= learning_rate * (gradients[f'dW{i}'] + lambd * parameters[f'W{i}'] + lambd * np.sign(parameters[f'W{i}']))
        parameters[f'b{i}'] -= learning_rate * gradients[f'db{i}']
    return parameters

##################################
# Implementing neural network model training
# with ElasticNet regularization
##################################

##################################
# Initializing training parameters
##################################
np.random.seed(88888)
parameters = initialize_parameters(input_size, hidden_sizes, output_size)

##################################
# Initializing regularization parameters
##################################
lambd = 0.01

##################################
# Creating lists to store cost and accuracy for plotting
##################################
costs = []
accuracies = []

##################################
# Creating lists to store weights for plotting
##################################
weight_history = {f'W{i}': [] for i in range(1, len(hidden_sizes) + 2)}

##################################
# Training a neural network model
# using ElasticNet regularization
##################################
for i in range(iterations):
    # Implementing forward propagation
    cache = forward_propagation(matrix_x_values, parameters)
    Y_hat = cache[f'A{len(parameters) // 2}']
    
    # Computing cost and accuracy
    cost = compute_cost(y_values, Y_hat)
    accuracy = np.mean(np.argmax(y_values, axis=1) == np.argmax(Y_hat, axis=1))
    
    # Implementing backward propagation
    gradients = backward_propagation(matrix_x_values, y_values, parameters, cache)
    
    # Updating model parameter values
    parameters = update_parameters_elastic(parameters, gradients, learning_rate, lambd)
    
    # Recording model weight values
    for j in range(1, len(hidden_sizes) + 2):
        weight_history[f'W{j}'].append(parameters[f'W{j}'].copy())
    
    # Recording cost and accuracy values
    costs.append(cost)
    accuracies.append(accuracy)
    
    # Printing cost and accuracy every 100 iterations
    if i % 100 == 0:
        print(f"Iteration {i}: Cost = {cost}, Accuracy = {accuracy}")

Iteration 0: Cost = 0.6655730513519705, Accuracy = 0.7484662576687117
Iteration 100: Cost = 0.44718825489399394, Accuracy = 0.7484662576687117
Iteration 200: Cost = 0.43400594664030084, Accuracy = 0.7484662576687117
Iteration 300: Cost = 0.4257228664966347, Accuracy = 0.7484662576687117
Iteration 400: Cost = 0.4190516925903125, Accuracy = 0.7484662576687117
Iteration 500: Cost = 0.4134013292735411, Accuracy = 0.7484662576687117
Iteration 600: Cost = 0.4083585633198166, Accuracy = 0.7484662576687117
Iteration 700: Cost = 0.4037088548290823, Accuracy = 0.7484662576687117
Iteration 800: Cost = 0.39966601427718335, Accuracy = 0.7484662576687117
Iteration 900: Cost = 0.39594057532140947, Accuracy = 0.7484662576687117
Iteration 1000: Cost = 0.3925139483320749, Accuracy = 0.7484662576687117
Iteration 1100: Cost = 0.38909789089024854, Accuracy = 0.7484662576687117
Iteration 1200: Cost = 0.3859648376461514, Accuracy = 0.7484662576687117
Iteration 1300: Cost = 0.3829541532661204, Accuracy = 0.7791411042944786
Iteration 1400: Cost = 0.38006500342073213, Accuracy = 0.7791411042944786
Iteration 1500: Cost = 0.37742536107438396, Accuracy = 0.7791411042944786
Iteration 1600: Cost = 0.3748963254187374, Accuracy = 0.803680981595092
Iteration 1700: Cost = 0.37241872043683283, Accuracy = 0.803680981595092
Iteration 1800: Cost = 0.36997305640633676, Accuracy = 0.8098159509202454
Iteration 1900: Cost = 0.3675035672115191, Accuracy = 0.8159509202453987
Iteration 2000: Cost = 0.3648282909497678, Accuracy = 0.7975460122699386
Iteration 2100: Cost = 0.3620304435865962, Accuracy = 0.8159509202453987
Iteration 2200: Cost = 0.3591947317998126, Accuracy = 0.8588957055214724
Iteration 2300: Cost = 0.35638431976187446, Accuracy = 0.8711656441717791
Iteration 2400: Cost = 0.3532542424136821, Accuracy = 0.8834355828220859
Iteration 2500: Cost = 0.3500658151702217, Accuracy = 0.8588957055214724
Iteration 2600: Cost = 0.3463750092649884, Accuracy = 0.8650306748466258
Iteration 2700: Cost = 0.34236941843663377, Accuracy = 0.8650306748466258
Iteration 2800: Cost = 0.3383693028183223, Accuracy = 0.8650306748466258
Iteration 2900: Cost = 0.3334202259570756, Accuracy = 0.8650306748466258
Iteration 3000: Cost = 0.3214396170399764, Accuracy = 0.8711656441717791
Iteration 3100: Cost = 0.3091981137736961, Accuracy = 0.8773006134969326
Iteration 3200: Cost = 0.29804017477679723, Accuracy = 0.8773006134969326
Iteration 3300: Cost = 0.2871442159702661, Accuracy = 0.8773006134969326
Iteration 3400: Cost = 0.2762105091956162, Accuracy = 0.8773006134969326
Iteration 3500: Cost = 0.2649560804833031, Accuracy = 0.8834355828220859
Iteration 3600: Cost = 0.253382562557902, Accuracy = 0.9263803680981595
Iteration 3700: Cost = 0.2420466877004572, Accuracy = 0.9263803680981595
Iteration 3800: Cost = 0.23020510840046451, Accuracy = 0.9263803680981595
Iteration 3900: Cost = 0.21841095478633638, Accuracy = 0.9325153374233128
Iteration 4000: Cost = 0.20712521537328352, Accuracy = 0.9447852760736196
Iteration 4100: Cost = 0.1955794071336753, Accuracy = 0.950920245398773
Iteration 4200: Cost = 0.18458063395853705, Accuracy = 0.950920245398773
Iteration 4300: Cost = 0.17151488911578616, Accuracy = 0.9570552147239264
Iteration 4400: Cost = 0.15857163162598373, Accuracy = 0.9693251533742331
Iteration 4500: Cost = 0.14809318432727417, Accuracy = 0.9693251533742331
Iteration 4600: Cost = 0.13931460366074877, Accuracy = 0.9693251533742331
Iteration 4700: Cost = 0.13045608405568113, Accuracy = 0.9693251533742331
Iteration 4800: Cost = 0.1222876074524626, Accuracy = 0.9754601226993865
Iteration 4900: Cost = 0.1161674176708189, Accuracy = 0.9693251533742331
Iteration 5000: Cost = 0.10962180845769681, Accuracy = 0.9693251533742331

##################################
# Plotting cost and accuracy profiles
##################################
plt.figure(figsize=(15, 4.75))
plt.subplot(1, 2, 1)
plt.plot(range(iterations), costs)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.title('ElasticNet Regularization: Cost Function by Iteration')

plt.subplot(1, 2, 2)
plt.plot(range(iterations), accuracies)
plt.ylim([0,1])
plt.xlabel('Iterations')
plt.ylabel('Accuracy')
plt.title('ElasticNet Regularization: Classification Accuracy by Iteration')

plt.show()

##################################
# Plotting model weight profiles
##################################
num_layers = len(hidden_sizes) + 1
plt.figure(figsize=(15, 10))
for i in range(1, num_layers + 1):
    plt.subplot(2, num_layers // 2, i)
    plt.plot(range(iterations), [np.max(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Maximum Weight")
    plt.plot(range(iterations), [np.mean(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Mean Weight")
    plt.plot(range(iterations), [np.min(np.abs(weights)) for weights in weight_history[f'W{i}']], label="Minimum Weight")
    plt.ylim([-1,5])
    plt.legend(loc="upper right")
    plt.xlabel('Iterations')
    plt.ylabel('Average Absolute Weight')
    plt.title(f'ElasticNet Regularization: Layer {i} Weights by Iteration')

plt.show()

##################################
# Gathering the final accuracy, cost 
# and mean layer weight values for 
# ElasticNet Regularization
##################################
ENR_metrics = pd.DataFrame(["ACCURACY","LOSS","LAYER 1 MEAN WEIGHT","LAYER 2 MEAN WEIGHT","LAYER 3 MEAN WEIGHT","LAYER 4 MEAN WEIGHT"])
ENR_values = pd.DataFrame([accuracies[-1],
                           costs[-1],
                           [np.mean(np.abs(weights)) for weights in weight_history['W1']][-1],
                           [np.mean(np.abs(weights)) for weights in weight_history['W2']][-1],
                           [np.mean(np.abs(weights)) for weights in weight_history['W3']][-1],
                           [np.mean(np.abs(weights)) for weights in weight_history['W4']][-1]])
ENR_method = pd.DataFrame(["ElasticNet Regularization"]*6)
ENR_summary = pd.concat([ENR_metrics, 
                         ENR_values,
                         ENR_method], axis=1)
ENR_summary.columns = ['Metric', 'Value', 'Method']
ENR_summary.reset_index(inplace=True, drop=True)
display(ENR_summary)

	Metric	Value	Method
0	ACCURACY	0.969325	ElasticNet Regularization
1	LOSS	0.109622	ElasticNet Regularization
2	LAYER 1 MEAN WEIGHT	0.275963	ElasticNet Regularization
3	LAYER 2 MEAN WEIGHT	0.374856	ElasticNet Regularization
4	LAYER 3 MEAN WEIGHT	0.371444	ElasticNet Regularization
5	LAYER 4 MEAN WEIGHT	0.596937	ElasticNet Regularization

1.7. Consolidated Findings

2. Summary

3. References

• [Book] Deep Learning: A Visual Approach by Andrew Glassner
• [Book] Deep Learning with Python by François Chollet
• [Book] The Elements of Statistical Learning by Trevor Hastie, Robert Tibshirani and Jerome Friedman
• [Book] Data Preparation for Machine Learning: Data Cleaning, Feature Selection, and Data Transforms in Python by Jason Brownlee
• [Book] Feature Engineering and Selection: A Practical Approach for Predictive Models by Max Kuhn and Kjell Johnson
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• [Book] Applied Predictive Modeling by Max Kuhn and Kjell Johnson
• [Book] Data Mining: Practical Machine Learning Tools and Techniques by Ian Witten, Eibe Frank, Mark Hall and Christopher Pal
• [Book] Data Cleaning by Ihab Ilyas and Xu Chu
• [Book] Data Wrangling with Python by Jacqueline Kazil and Katharine Jarmul
• [Book] Regression Modeling Strategies by Frank Harrell
• [Python Library API] NumPy by NumPy Team
• [Python Library API] pandas by Pandas Team
• [Python Library API] seaborn by Seaborn Team
• [Python Library API] matplotlib.pyplot by MatPlotLib Team
• [Python Library API] itertools by Python Team
• [Python Library API] operator by Python Team
• [Python Library API] sklearn.experimental by Scikit-Learn Team
• [Python Library API] sklearn.impute by Scikit-Learn Team
• [Python Library API] sklearn.linear_model by Scikit-Learn Team
• [Python Library API] sklearn.preprocessing by Scikit-Learn Team
• [Python Library API] scipy by SciPy Team
• [Article] Step-by-Step Exploratory Data Analysis (EDA) using Python%20with,distributions%20using%20Python%20programming%20language.) by Malamahadevan Mahadevan (Analytics Vidhya)
• [Article] Exploratory Data Analysis in Python — A Step-by-Step Process by Andrea D'Agostino (Towards Data Science)
• [Article] Exploratory Data Analysis with Python by Douglas Rocha (Medium)
• [Article] 4 Ways to Automate Exploratory Data Analysis (EDA) in Python by Abdishakur Hassan (BuiltIn)
• [Article] 10 Things To Do When Conducting Your Exploratory Data Analysis (EDA) by Alifia Harmadi (Medium)
• [Article] How to Handle Missing Data with Python by Jason Brownlee (Machine Learning Mastery)
• [Article] Statistical Imputation for Missing Values in Machine Learning by Jason Brownlee (Machine Learning Mastery)
• [Article] Imputing Missing Data with Simple and Advanced Techniques by Idil Ismiguzel (Towards Data Science)
• [Article] Missing Data Imputation Approaches | How to handle missing values in Python by Selva Prabhakaran (Machine Learning +)
• [Article] Master The Skills Of Missing Data Imputation Techniques In Python(2022) And Be Successful by Mrinal Walia (Analytics Vidhya)
• [Article] How to Preprocess Data in Python by Afroz Chakure (BuiltIn)
• [Article] Easy Guide To Data Preprocessing In Python by Ahmad Anis (KDNuggets)
• [Article] Data Preprocessing in Python by Tarun Gupta (Towards Data Science)
• [Article] Data Preprocessing using Python by Suneet Jain (Medium)
• [Article] Data Preprocessing in Python by Abonia Sojasingarayar (Medium)
• [Article] Data Preprocessing in Python by Afroz Chakure (Medium)
• [Article] Detecting and Treating Outliers | Treating the Odd One Out! by Harika Bonthu (Analytics Vidhya)
• [Article] Outlier Treatment with Python by Sangita Yemulwar (Analytics Vidhya)
• [Article] A Guide to Outlier Detection in Python by Sadrach Pierre (BuiltIn)
• [Article] How To Find Outliers in Data Using Python (and How To Handle Them) by Eric Kleppen (Career Foundry)
• [Article] Statistics in Python — Collinearity and Multicollinearity by Wei-Meng Lee (Towards Data Science)
• [Article] Understanding Multicollinearity and How to Detect it in Python by Terence Shin (Towards Data Science)
• [Article] A Python Library to Remove Collinearity by Gianluca Malato (Your Data Teacher)
• [Article] 8 Best Data Transformation in Pandas by Tirendaz AI (Medium)
• [Article] Data Transformation Techniques with Python: Elevate Your Data Game! by Siddharth Verma (Medium)
• [Article] Data Scaling with Python by Benjamin Obi Tayo (KDNuggets)
• [Article] How to Use StandardScaler and MinMaxScaler Transforms in Python by Jason Brownlee (Machine Learning Mastery)
• [Article] Feature Engineering: Scaling, Normalization, and Standardization by Aniruddha Bhandari (Analytics Vidhya)
• [Article] How to Normalize Data Using scikit-learn in Python by Jayant Verma (Digital Ocean)
• [Article] What are Categorical Data Encoding Methods | Binary Encoding by Shipra Saxena (Analytics Vidhya)
• [Article] Guide to Encoding Categorical Values in Python by Chris Moffitt (Practical Business Python)
• [Article] Categorical Data Encoding Techniques in Python: A Complete Guide by Soumen Atta (Medium)
• [Article] Categorical Feature Encoding Techniques by Tara Boyle (Medium)
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• [Article] Hypothesis Testing with Python: Step by Step Hands-On Tutorial with Practical Examples by Ece Işık Polat (Towards Data Science)
• [Article] 17 Statistical Hypothesis Tests in Python (Cheat Sheet) by Jason Brownlee (Machine Learning Mastery)
• [Article] A Step-by-Step Guide to Hypothesis Testing in Python using Scipy by Gabriel Rennó (Medium)
• [Article] How Does Backpropagation in a Neural Network Work? by Anas Al-Masri (Builtin)
• [Article] A Step by Step Backpropagation Example by Matt Mazur (MattMazur.Com)
• [Article] Understanding Backpropagation by Brent Scarff (Towards Data Science)
• [Article] Understanding Backpropagation Algorithm by Simeon Kostadinov (Towards Data Science)
• [Article] A Comprehensive Guide to the Backpropagation Algorithm in Neural Networks by Ahmed Gad (Neptune.AI)
• [Article] Backpropagation by John McGonagle, George Shaikouski and Christopher Williams (Brilliant)
• [Article] Backpropagation in Neural Networks by Inna Logunova (Serokell.IO)
• [Article] Backpropagation Concept Explained in 5 Levels of Difficulty by Devashish Sood (Medium)
• [Article] BackProp Explainer by Donny Bertucci (GitHub)
• [Article] Backpropagation Algorithm in Neural Network and Machine Learning by Intellipaat Team
• [Article] Understanding Backpropagation in Neural Networks by Tech-AI-Math Team
• [Article] Backpropagation Neural Network using Python by Avinash Navlani (Machine Learning Geek)
• [Article] Back Propagation in Neural Network: Machine Learning Algorithm by Daniel Johnson (Guru99)
• [Article] What is Backpropagation? by Thomas Wood (DeepAI.Org)
• [Article] Activation Functions in Neural Networks [12 Types & Use Cases] by Pragati Baheti (V7.Com)
• [Article] Activation Functions in Neural Networks by Sagar Sharma (Towards Data Science)
• [Article] A Practical Comparison of Activation Functions by Danny Denenberg (Medium)
• [Article] What is an Activation Function? A Complete Guide by Petru Potrimba (RoboFlow.Com)
• [Article] L1, L2 and Elastic Net Regularization in Neural Networks (A Deep Dive) by Francesco Franco (AI.Mind.SO)
• [Article] Regularization in Neural Networks by Bala Priya (PineCone.IO)
• [Article] Regularization in Deep Learning — L1, L2, and Dropout by Artem Oppermann (Towards Data Science)
• [Article] Regularization Techniques for Training Deep Neural Networks bySergios Karagiannakos (TheAISummer.Com)
• [Article] What is Regularization? by IBM Team
• [Article] Regularization — Understanding L1 and L2 regularization for Deep Learning by Ujwal Tewari (Medium)
• [Article] A Guide to Regularization in Python by Sadrach Pierre (BuiltIn)
• [Article] Regularization: Make your Machine Learning Algorithms Learn, not Memorize by Anand Borad (EInfoChips.Com)
• [Article] Regularization In Neural Networks by Egor Howell (Towards Data Science)
• [Article] Elastic Net Regression : The Best of L1 and L2 Norm Penalties by Aditya Kakde (Medium)
• [Article] L1 And L2 Regularization Explained, When To Use Them & Practical How To Examples by Neri Van Otten (SpotIntelligence.Com)
• [Article] Elastic Net Regression —Combined Features of L1 and L2 regularization by Abhishek Jain (Medium)
• [Article] The Choice of Regularization: Ridge, Lasso and Elastic Net Regression by Rukshan Pramoditha (Towards Data Science)
• [Article] Regularization Techniques in Keras Neural Networks by Okan Yenigün (AIMind.SO)
• [Article] Courage to learn ML: Demystifying L1 & L2 Regularization (Part 1) by Amy Ma (Towards Data Science)
• [Article] Courage to learn ML: Demystifying L1 & L2 Regularization (Part 2) by Amy Ma (Towards Data Science)
• [Article] Courage to learn ML: Demystifying L1 & L2 Regularization (Part 3) by Amy Ma (Towards Data Science)
• [Article] L1, L2 and Elastic Net Regularization in Neural Networks (A Deep Dive) by Ramsey Elbasheer
• [Article] Regularization for Neural Networks with Framingham Case Study by Rachel Draelos (GlassBoxMedicine.Com)
• [Article] Regularization in Machine Learning by Prashant Gupta (Towards Data Science)
• [Article] Regularization Techniques in Deep Learning: Ultimate Guidebook by Turing Team
• [Article] Regularization by ML-Cheatsheet Team
• [Article] Explaining L1 and L2 Regularization in Machine Learning by Fernando Jean Dijkinga (Medium)
• [Article] Deep Learning Best Practices: Regularization Techniques for Better Neural Network Performance by Niranjan Kumar (HeartBeat.Comet.ML)
• [Publication] Data Quality for Machine Learning Tasks by Nitin Gupta, Shashank Mujumdar, Hima Patel, Satoshi Masuda, Naveen Panwar, Sambaran Bandyopadhyay, Sameep Mehta, Shanmukha Guttula, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’21: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining)
• [Publication] Overview and Importance of Data Quality for Machine Learning Tasks by Abhinav Jain, Hima Patel, Lokesh Nagalapatti, Nitin Gupta, Sameep Mehta, Shanmukha Guttula, Shashank Mujumdar, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)
• [Publication] Multiple Imputation of Discrete and Continuous Data by Fully Conditional Specification by Stef van Buuren (Statistical Methods in Medical Research)
• [Publication] Mathematical Contributions to the Theory of Evolution: Regression, Heredity and Panmixia by Karl Pearson (Royal Society)
• [Publication] A New Family of Power Transformations to Improve Normality or Symmetry by In-Kwon Yeo and Richard Johnson (Biometrika)
• [Course] IBM Data Analyst Professional Certificate by IBM Team (Coursera)
• [Course] IBM Data Science Professional Certificate by IBM Team (Coursera)
• [Course] IBM Machine Learning Professional Certificate by IBM Team (Coursera)
• [Course] Machine Learning Specialization Certificate by DeepLearning.AI Team (Coursera)

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